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Mirrors > Home > ILE Home > Th. List > dcned | GIF version |
Description: Decidable equality implies decidable negated equality. (Contributed by Jim Kingdon, 3-May-2020.) |
Ref | Expression |
---|---|
dcned.eq | ⊢ (𝜑 → DECID 𝐴 = 𝐵) |
Ref | Expression |
---|---|
dcned | ⊢ (𝜑 → DECID 𝐴 ≠ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dcned.eq | . . 3 ⊢ (𝜑 → DECID 𝐴 = 𝐵) | |
2 | dcn 842 | . . 3 ⊢ (DECID 𝐴 = 𝐵 → DECID ¬ 𝐴 = 𝐵) | |
3 | 1, 2 | syl 14 | . 2 ⊢ (𝜑 → DECID ¬ 𝐴 = 𝐵) |
4 | df-ne 2348 | . . 3 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵) | |
5 | 4 | dcbii 840 | . 2 ⊢ (DECID 𝐴 ≠ 𝐵 ↔ DECID ¬ 𝐴 = 𝐵) |
6 | 3, 5 | sylibr 134 | 1 ⊢ (𝜑 → DECID 𝐴 ≠ 𝐵) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 DECID wdc 834 = wceq 1353 ≠ wne 2347 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-ne 2348 |
This theorem is referenced by: nn0n0n1ge2b 9308 algcvgblem 12019 |
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